A Hydrothermal Problem with Non-smooth Lagrangian

This paper deals with the optimization of a hydrothermal problem that considers a non-smooth Lagrangian L(t, z, z′). We consider a general case where the functions Lz′ (t, ·, ·) and Lz(t, ·, ·) are discontinuous in {(t, z, z′)/z′ = φ(t, z)}, which is the borderline point between two power generation zones. This situation arises in problems of optimization of hydrothermal systems where the thermal plant input-output curve considers the shape of the cost curve in the neighborhood of the valve points. The problem shall be formulated in the framework of nonsmooth analysis, using the generalized (or Clarke’s) gradient. We shall obtain a necessary minimum condition and we shall generalize the known result (smooth transition) that the derivative of the minimum presents a constancy interval. Finally, we shall present an example.